Although many excellent papers and some specialised monographs on different aspects of design theory had been published, prior to 1985 (when the first edition of this book appeared) there was no comprehensive monograph on the field. Thus it was our plan to cover the main concepts and ideas of modern design theory without being encyclopedic, but including a deeper study of several representative topics. As it turned out, these aims (which required a rather long book) seem to have been met by the first edition which has been quoted extensively in the research literature.

A first draft of this book was obtained by merging different manuscripts of the authors. The first edition then grew from an iterative process of rewriting in which all the authors contributed their ideas to each of the parts. For reasons of time constraints and shifting research interests, the major part of the revision now presented has been done by the second author, drawing on his previous update Jungnickel (1989a) and surveys Jungnickel (1990a, 1992a); in particular, this holds for Chapter VI. Of course, there has still been considerable help from his co-authors (with Chapter XIII being the first author's contribution), but nevertheless he is willing to accept most of the blame for the mistakes introduced during the process of revision.

Algebraic combinatorics involves the use of techniques from algebra, topology and geometry in the solution of combinatorial problems, or the use of combinatorial methods to attack problems in these areas. Problems amenable to the methods of algebraic combinatorics arise in these or other areas of mathematics, or from diverse parts of applied mathematics. Because of this interplay with many fields of mathematics, algebraic combinatorics is an area in which a wide variety of ideas and methods come together.

During 1996-97 MSRI held a full academic year program on Combinatorics, with special emphasis on algebraic combinatorics and its connections with other branches of mathematics, such as algebraic geometry, topology, commutative algebra, representation theory, and convex geometry. Different periods of the year were devoted to research in enumeration, extremal questions, geometric combinatorics and representation theory.

The rich combinatorial problems arising from the study of these various areas are the subject of this book, which represents work done or presented at seminars during the program. It contains contributions on matroid bundles, combinatorial representation theory, lattice points in polyhedra, bilinear forms, combinatorial differential topology and geometry, Macdonald polynomials and geometry, enumeration of matchings, the generalized Baues problem, and Littlewood- Richardson semigroups. These expository articles, written by some of the most respected researchers in the field, present the state-of-the-art to graduate students and researchers in combinatorics as well as algebra, geometry, and topology.

1. The Varchenko B Matrices

Let A = ﹛H1,…, Hl﹜ be an arrangement of hyperplanes in ℝn and let r(A) = ﹛R1,…, Rm﹜ denote the set of regions in the complement of the union of .A. Let L(A) denote the collection of intersections of hyperplanes in A. Included in L(A) is ℝ n, which we think of as the intersection of the empty set of hyperplanes. We order the elements of L(A) by reverse inclusion thus making it into a poset. It is well known that this poset is a meet semilattice and is a geometric lattice if the arrangement is central. We will abbreviate L(A) to L when the arrangement is clear.

This article is based on my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. I thank the organizers (Sergey Fomin, Curtis Greene, Phil Hanlon and Sheila Sundaram) for bringing together a group of outstanding combinatorialists and for giving me a chance to bring to their attention some of the problems that I find very exciting and beautiful. In preparing the note for this volume (October 1998), I made a few small changes in the original version [Zelevinsky 1997], and added in the end a brief (and undoubtedly incomplete) account of some exciting progress achieved since April 1997. I am grateful to the referee for helpful suggestions.

Theorem 1. LRr is a finitely generated subsemigroup of the additive semigroup Pr3 ⊂ ℤ3r. This is a special case of a much more general result well known to the experts in invariant theory. A short proof (valid for any reductive group instead of GLr(ℂ)) can be found in [Elashvili 1992]; A. Elashvili attributes this proof to M. Brion and F. Knop. The semigroup property also follows at once from “polyhedral” expressions for that will be discussed later (see Theorem 5 and below).

Introduction

A variety of questions in combinatorics lead one to the task of analyzing a simplicial complex, or a more general cell complex. For example, a standard approach to investigating the structure of a partially ordered set is to instead study the topology of the associated order complex. However, there are few general techniques to aid in this investigation. On the other hand, the subjects of differential topology and differential geometry are devoted to precisely this sort of problem, except that the topological spaces in question are smooth manifolds, rather than combinatorial complexes. These are classical subjects, and numerous very general and powerful techniques have been developed and studied over the recent decades.

A smooth manifold is, loosely speaking, a topological space on which one has a well-defined notion of a derivative. One can then use calculus to study the space. I have recently found ways of adapting some techniques from differential topology and differential geometry to the study of combinatorial spaces. Perhaps surprisingly, many of the standard ingredients of differential topology and differential geometry have combinatorial analogues.